linear programming 09 linear programming

161Linear Programming

MATHEMATICS – MHT-CET Himalaya Publication Pvt. Ltd.


Syllabus

Linear Programming Problem General Form of

Linear Programming Problem Different Types of

Linear Programming Problems Solution of LPP

by Graphical Method

The term 'programming' means 'planning' and it

refers to a particular plan of action from amongst

several alternatives for maximising or minimising

a function under given restrictions as maximising

profit or minimising cost, etc. The term 'linear'

means that 'all inequations' or 'equations used'

and the function to be maximised or minimised

are linear. Thus, linear programming is a technique

for resource utilisation.

Linear Programming Problem

A linear programming problem may be defined as

the problem of maximising or minimising a linear

function subject to linear constraints. The

constraints may be equalities or inequalities.

Important Terms Related to LPP

There are many terms related to a linear

programming problem. Definition of these terms

are given below:

Constraints

The linear inequations or inequalities or restrictions

on the variables of a linear programming problem

are called constraints. The conditions x > 0, y > 0

are called non-negative restrictions.

Optimisation Problem

Aproblem which seeks to maximise or mini mise

a linear function subject to certain constraints as

determined by a set of linear inequalities is called

an optimisation problem. Linear programming

problems are special type of optimisation

problems.

Objective Function

A linear function of two or more variables which

has to be maximised or minimised under the given

restrictions is called an objective function.The

variables used in the objective function are called

decision variables.

Optimal Value

The maximum or minimum value of an objective

function is known as the optimal value of LPP.

Feasible Region

The common region determined by all the

constraints including non-negative constraints

x,y > 0 of a linear programming problem is called

the feasible region or solution region. Each point

in this region represents a feasible choice. The

region other than feasible region is called an

infeasible region.

Bounded and Unbounded Region

A feasible region of a system of linear inequalities

is said to be bounded, if it can be enclosed within

a circle. Otherwise, it is said to be unbounded

region i.e. the feasible region does extend

indefinitely in any direction.

Feasible Solution

Points within and on the boundary of the feasible

region represent feasible solution of the

constraints. Any point outside the feasible region

is called an infeasible solution.

Optimal Feasible Solution

A feasible solution at which the objective function

has optimal value (maximum or minimum) is

called the optimal solution or optimal feasible

solution of the linear programming problem.

Optimisation Technique

The process of obtaining the optimal solution of

the linear programming problem is called

optimisation technique.

Mathematical Form of LPP

The general mathematical form of a linear

programming problem may be written as follows:

Objective function, z = c1 x + c

2y

Subject to constraints are alx + b

1y

2 < d

1,

a2x + b

2y < d etc.

and non-negative restrictions are x > 0, y > 0.

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DGT MH –CET 12th MATHEMATICS Study Material 1



General Form of Linear Programming

Problem

The mathematical statement of the general form

of a linear programming problem (abbreviated

LPP) may be written as follows:

Optimise (maximise or minimise)

Z = c1x

1 + c

zx

z + ... + c

nx

n

subject to constraints are

a11

x1 + a

12x

2 +.... + a

1nx

n (<, = >} b

1

a21

x1 + a

22x

2 + ... + a

2nx

n (<, = >} b

2 ...(i)

and am1x

1 + a

m2x

2 + ...+ a

mnx

n ((<, = >} b

m

where, x1,x

2 ..... ,x

n > 0, ...(ii)

i. x1, x

2, ..., Xn are the variable who e values we

wish to determine and are called the decision or

structural variables.

ii. The linear function Z which is to be maximised

or minimised is called the objective function of

the general LPP.

iii. The inequalities (i) are called the constraints of

the general LPP.

iv. The set of inequalities (ii) is known as the et of

non-negative restrictions to the general LPP.

v. The constant cj (j=1,2, ..., n) repre en the

contribution (profit or cost) to the objective

function of the jth variable.

vi. The coefficient aij (i = 1,2, ...,m; j = 1,2 ..., n) are

referred to as the technological or subtitution

coefficients.

vii. bi (i = 1, 2 ...., m) is the constant repre enting the

requirement or availability of the jth constraint.

viii. The expression (< =, >) means that onI one of

the relationship is the set (<, =, > ) should hold

for a particular constraint

Example 1

A firm has to transport 1200 packages using large

vans which can carry 200 packages each and small

van which can take 80 packages each. The cost

for engaging each large van is Rs. 400 and each

small van is Rs. 200. Not more than Rs. 3000 is

to be spent on the job and the number of large

vans cannot exceed the number of small vans.

Formulation of this problem as a LPP gi en that

the objective is to minimise cost is

a. minirni e = Z = 200y + 400x,

subject to constraints

5x + 2y > 30,

2x + y < 15,

x < y,

x, y, > 0

b. minimise z = 400x + 200y, subject to constraints

5x + 2y < 5 : 30

2x + y < 5 : 15,

x < y,

x, y > 0

c. minirniseZ = 200Y + 400x, subject to constraints

5x + 2y < 5 : 30

2x + y : 5 < 15,

x < y,

x,y > O

d. rninimise Z = 400x + 200y, subject to constraints

5x+ 2y < 5:30

2x+ y : 5 < 15,

x < y,

x,y < 0

Sol (b) Let the firm has x number of large vans and y

number of small vans. From the given information,

we have following corresponding con traint table.

Large vans (x) Small vans(y) Maximum / Minimum

Packages 200 80 1200

Cost 400 200 3000

Thus, we see that objective function for minimum

cost is

200 = 400x + 80y > 1200y.

Subject to con traints

200x + 80y > 1200 [package constraint)

5x + 2y > 30 ...(i)

and 400x + 200y < 3000 [cost constraint)

2x + y < 15 ... (ii)

and x < y [van constraint) ... (iii)

and x > 0, y > 0 [non-negativeconstraints).. (iv)

Thus, required LPP to minimise cost is

mini mise Z = 400x + 200y, subject to constraints

5x + 2y > 30,

2x + y < 15,

x < y,

x > 0, y > 0





Different Types of Linear Programming

Problems

Some important types of linear programming

problems are given below:

To solve such problems, we first formulate the

mathematical form of LPP and then solve by using

corner point method, which will be discussed later.

Diet Problems

In diet problems, we have to determine the amount

of different kinds of constituents/nutrients which

should be included in a diet so as to minimise the

cost of the desired diet such that it contains a

certain minimum amount of each constituent/

nutrient.

• Example 2

(Diet problem) Adietician wishes to mix two types

of foods f1 and f

2 in such a way that the vitamin

contents of the mixture contain atleast 6 units of

vitamin A and 8 units of vitamin f1. Food II

contains 2 units/kg of vitamin A and 3 units/kg of

vitamin B while food 12 contains 3 units/kg of

vitamin A and 2 units/kg of vitamin B. Food f2

cost Rs. 50 per kg and food f2 cost Rs. 75 per kg.

Formulate the problem as an LPP to minimise

the cost of mixture.

a. min. Z = 50x + 75y

b. min. Z = 50x + 75y

subject to constraints subject to constraints,

2x + 3y > 6 2x+3y:5:6

3x + 2y > 8 3x + 2y < 8

and x, y > 0 and x, y > 0

c.min.Z=50x+75y d.Noneofthese

subject to constraints

2x + 3y > 8

3x + 2y < 6

and x, y > 0

Sol (a) Let the mixture contains food

f1 = x kg and food f

2 = y kg.

Here, cost of food 11 = Rs. 50 per kg and cost of

food 12 = Rs. 75 per kg and we have to minimise

the cost of mixture. So, the objective function is

to minimise, Z = 50x + 75y

Here, food f1

contains 2 units/kg and food f2

contains 3 units/kg of vitamin A. Also, mixture

contain atleast 6 units of vitamin A. So, first

constraint is 2x + 3y > 6.

Similarly, food 11 contains 3 units/kg and food f2

contains 2 units/kg of vitamin B. Also, mixture

contain atleast 8 units of vitamin B. So, second

constraint is 3x + 2y > 8.

Also, foods f1 and f

2 cannot be negative, so

x > 0, y – 0.

Thus,

Formation of this LPP is, objective function,

Minimum, Z = SOx + 7sy

subject to constraints, 2x + 3y > 6

3x + 2y >8 and x,y > 0

Manufacturing Problems

In manufacturing problems, we have to determine

the number of units of different products which

should be produced and sold by a firm, when

each product requires a fixed manpower, machine

hours, labour hour per unit of product, warehouse

space per unit of the output, etc. in order to make

maximum profit.

Example 3

A manufacturer considers that men and women

workers are equally efficient and so he pays them

at the same rate. He has 30 and 17 units of workers

(male and female) and capital respectively, which

he uses to produce two types of goods A and B.

To produce 1unit of A, 2 workers and 3 units of

capital are required while 3 workers and 1 unit of

capital is required to produce 1 unit of B. If A and

B are priced at Rs.100 and Rs. 120 per unit

respectively. Form the above as an LPP to

maximise the revenue.

a. maximise Z=100x+ 120y, subject to constraints

2x +3y < 30,

3x + y < 17,

x,y > o

b. maximise Z = 80x + 120y, subject to constraints

5x+ 2y < 16

x + y < 15,

x < y,

x, y > 0

c. maximise z = 100x+ 80 y, subject to constraints

2x + 5y < 30

2x + y <15,

x < y

x,y > 0





d. maximise Z = 40x + 120y, subject to constraints

3x + 2y < 30

2x + y < 10,

x < y,

x,y > o

Sol (a) Let the manufacturer produces x units of goods

A and y units of goods B.

Now, formulate a table for given data

A B Required capacity

Workers 2 3 30

Capital 3 1 17

Profit function (z) 100 120

The required LPP becomes,

Maximise z =100x + 120y

Subject to constraints are 2x + 3y < 30 ....(i)

3x + y < 17 ...(ii)

and x, y > 0 .... (iii)

which is required formulation of the given

manufacturing problem.

Transportation Problems

In transportation problems, we have to determine

a transportation schedule in order to find the

minimum cost of transporting a product from

plants/factories situated at different locations to

different markets.

Example 4

There is a factory located at each of the two places

P and Q.

From these locations, certain commodity is

delivered to each of the three depots situated atA,

Band C. The weekly requirements of the depots

are respectively 5, 5 and 4 units of the commodity

while the production capacity of the factory at P

and Q are respectively, 8 and 6 units. The cost of

transportation per unit is given below:

To/From Cost (in Rs')

A B C

P 16 10 15

Q 10 12 10

Formulate the above LPP mathematically such

that the transportation cost is minimum.

a. minimise Z = x –7y+190, subject to constraints·

5x + 2y > 30,

2x + y < 15,

x < y,

x, y < O

b. minimise Z= x–7y + 190, subject to constraints

x + y < 8,

x + y > 4,

x < 5, y < 5

x, y > 0

c. minimise Z =x–7Y + 190, subject to constraints

x + y < 4

x + y > 4,

x, < 5, y > 5

x,y > 0

d. minimise Z=x – 7y + 190, subject to constraints

x + y < 4,

2x + 3y > 6,

x < 5, y < 5,

x,y > 0

Sol (b) Let the unit supplied from P to A be x and the

unit supplied form P to B be y.

Then, the unit supplied from P to C be 8 - (x + y)

and the unit supplied from Q to A be 5 – x and the

unit supplied from Q to B be 5 – y and the unit

supplied from Q to C be x + y – 4.

The objective function is to

Minimise Z = 16x + 10y + 15(8 – x – y) + 10

(5 – x) + 12(5 – y) + 10 (x + y – 4)

= 16 x + 10 + 120 –15x –15y + 50 – 10x

+ 60 –12y + 10x + 10y–40

= x – 7y + 190

Subject to the constraints are

x + y < 8 ... (i)

x + y > 4 ... (ii)





x < 5, y < 5 ...(iii)

and x,y > 0 ... (iv)

Which is required formulation of the given

transportation problem.

Solution of LPP by Graphical Method

There are two graphical methods:

i. Iso - profit or Iso-cost method

ii. Corner point method

Iso-profit or Iso-cost Method

If the objective function is of maximisation type,

then this initial objective line z1 = ax + by is called

iso-profit line. If the objective function is of

minimisation type, then this initial objective line

z1 = ax + by is called iso-cost line.

Working Rule to Find Solution of LPP by Iso-

profit or Iso-cost Method

i. Graph the constraints and identify the feasible

region.

ii. Draw the initial objective zl = ax + by which

passes through a point belonging to the feasible

region.

iii. If objective function is of maximisation type,

then move the iso-profit line z1=ax+ by parallel

to it self away from the origin, till the feasible

extreme points are located. This iso-profit line

gives maximum value of z.

iv. If the objective function is of minimisation type,

then move the iso-cost line parallel to itself

towards the origin till the feasible extreme points

are located. At this point(s) the objective

function will have minimum value.

Example 5

Solve the following linear programming problem

graphically: Maximize z = 50x + 15y

Subject to 5x + y < 100

x + y < 60

x, y > 0

a. 1200 b. 1240

c. 1260 d. 1250

Sol (d) We nrst convert the inequations into equations

to obtain the lines 5x + y = 100,

x + y = 60, x = 0 and y = 0.

The line 5x + y = 100 meets the coordinate axes

at A1 (20, 0) and B

1 (0,100). Join these points to

obtain the line 5x + y = 100.

The line x + y = 60 meets the coordinate axes at

A2 (60, 0) and B

2(0, 60). Join these points to

obtain the line x + y = 60.

Also, x = 0 is the Y-axis and y = 0 is X-axis.

The feasible region of the LPP is shaded in figure.

The coordinates of the corner points of the feasible

region OA1, PB

2 are 0 (0, 0), A

1(20, 0),

P(10, 50) and B2 (0, 60).

Now, we take a constant value, say 300 (i.e, 2

times the 1 cm of 50 and 15)for Z. Then,

300 = 50x + 15y

This line meets the coordinate axes at P1(6,0) and

Q1 (0,20). Join these points by a dotted line. Now,

move this line parallel to itself in the increasing

direction i.e. away from the origin. P2Q

2 and P

3Q

3

are such lines. Out ofthese lines locate a line which

is farthest from the origin and has at least one

point common to the feasible region.

Clearly, P3Q

3 is such line and it passes through

the vertex P(10, 15) the convex polygen OAl: PB

2.

Hence, x = 10 and y = 50 will give the maximum

value of Z. The maximum value of Z is given by

Z = 50 × 10 + 15 × 50 = 1250

Corner Point Method

Working Rule

i. Convert the inequality into equality .

ii. Find the solution set of the given constraints

(inequality) i.e. feasible region.

iii. Find the point of intersection of the lines.

iv. Find the corner points of the feasible region.

v. Find the value of objective function at each

corner point.

vi. Find the optimum solution i.e. maximum or

minimum value.





Example 6

The feasible region for a LPP is shown in the

following figure. Evaluate Z = 4x + yat each of

the corner points of this region. Then the minimum

value of Z, is (if it exists).

a. 9 b. 16

c. 3 d. one of these

Sol (c) From the shaded region, it is clear that

feasible region is unbounded with the corner points

A (4, 0), B (2,1) and C (0,3).

Also, we have Z = 4x + y.

[since, x+2y=4 and x + y = 3 Y = 1 and x = 2]

Comer points Corresponding value of Z

(4.0) 16

(2,1) 9

(0,3) 3 Minimum

Now, we see that 3 is the smallest value of Z at

the corner point (0,3). Note that here we see that,

the region is unbounded, therefore 3 may or may

not be the minimum value of Z.

To decide this issue, we graph the inequality

4x + y < 3 and check whether the resulting open

half plane has no point in common with feasible

region etherwlse. Z has no minimum value.

From the shown graph above, it is clear that there

is no point in common with feasible region and

hence Z has minimum value 3 at (0,3).

Special Cases of LPP

These special cases of LPP are given below:

i. Infinite Number of Optimal Solution

The LPP may have more than one solution

because of the nature of the objective function.

ii. Unbounded Solution

In some linear programming problem the solution

which are not bounded.

iii. Infeasible Solution

In some linear programming problem the feasible

solution does not exist.

Example 7

The linear programming problem

Maximise Z = x1 + x

2

Subject to constraints

x1 + 2x

2 < 2000, x

1 + x

2 < 1500, x

2 < 600

x1 > 0 has

a. no feasible solution

b. unique optimal solution

c. a finite number of optimal solutions

d. infinite number of optimal solutions

Sol (d) Given constraints are

x1 + 2x

2 < 2000, x

1 + x

2 < 1500, x

2 < 600 and

x1 > 0

Now, draw these inequalities, we have

The feasible region is OACDEO.

Given, Z = xl + x

2

AtO(0,0), Z = 0 + 0 = 0

At A (1500,0), Z = 1500 + 0 = 1500

At C (1000, 500), Z = 1000 + 500 = 1500

At D (800, 600), Z = 800 + 600 = 1400

At E (0,600), Z = 0 + 600 = 600

Here, Z is maximum on the segment AC.

Hence, there are infinite optimal solutions.





Formulation of LPP

1. Shaded region is represented by

a. 4x – 2y < 3 b. 4x – 2y < – 3

c. 4x – 2y > 3 d. 4x – 2y > 2 – 3

2. In case of a linear programming problem, feasible

region is always

a. a convex set

b. a concave set

c. a bounded convex set

d. a bounded concave set

3. The feasible region for the following constraints

L1 < 0, L

2 > 0, L

3

= 0 x > 0 , y > 0 in the diagram

shown is

a. area DHF b. area AHC

c. line segment EG d. line segment GI

e. line segment IC

4. A vertex of a feasible region by the linear

constraints 3x + 4y < 18, 2x + 3y > 3 and x, y > 0

is

a. (0, 2) b. (4.8, 0)

c. (0, 3) d. None of these

5. In linear programming problem, the linear function

Z subject to certain conditions determined by a

set of linear inequalities with variables as non-

negative, is

a. maximised only b. minimised only

c. Both (a) and (b) d. None of these

6. A problem which seeks to maximise or minimise

a linear function (say, two variables x and y)

subject to certain constraints as determined by a

set of linear inequalities is called alan

a. optimisation problem

b. functional problem

c. numerical problem

d. computer problem

7. The linear inequalities or equations or restrictions

on the variables of a linear programming problem

are called

a. linear relations b. constraints

c. functions d. objective functions

9. Consider the following statements

I. The term linear implies that all mathematical

relations used in the problem are linear relation

II. The term programming refers to the method

of determining a particular programme.

Choose the correct option.

a. Only I is true b. Only" is true

c. Both I and " are true d. Neither I nor" is true

10. A furniture dealer deals in only two items-tables

and chair. He has Rs. 50000 to invest and has

storage space of atmost 60 pieces. A table costs

Rs. 2500 and a chair Rs. 500. Then, the

constraints of the above problem are (where, x is

number of tables and y is number of chairs)

a. x > 0, y > 0 b. 5x + y < 100

c. x + y < 60 d. All of these

11. If a furniture dealer estimates that from the sale

of one table he can make a profit of Rs. 250 and

from the sale of one chair of a profit of Rs. 75

and if x is the number of chairs and y is the number

of tables, then its linear objective function is

a. Z = 75x + 250y b. Z = 75x + 25y

c. Z = 250x + 75y d. Z = 25x + 75y

12. A linear programming problem is one that is

concerned with finding the ... A. .. of a linear

function called ... B.... function of several

variables (say x and y), subject to the conditions

that the variables are ... C. .. and satisfy set of

linear inequalities called linear constraints.

Here, A, Band C are respectively

a. objective, optimal value, negative

b. optimal value, objective, negative

c. optimal value, objective, non-negative

d. objective, optimal value, non-negative

Exercise

(Topical Problems)





13. The conditions x > 0, y > 0 are called

a. restrictions only

b. negative restrictions

c. non-negative restrictions

d. None of these

13. Which of the following is a linear objective

function?

a. Z = ax + by b. Z < ax + by

c. Z > ax + by d. Z ax + by

14. The variables x and y in a linear programming

problem are called

a. decision variables b. linear variables

c. optimal variables d. None of these

15. The shaded region for the inequality x + 5 Y s 6 is

a. to the non-origin side of x + 5y = 6

b. to the either side of x + 5y = 6

c. to the origin side of x + 5y = 6

d. to the neither side of x + 5y = 6

16. Which of the following statements is false?

a. The feasible region is always a concave region

b. The maximum (or minimum) solution of the

objective function occurs at the vertex of the

feasible region

c. If two corner points produce the same

maximum (or minimum) value of the objective

function, then every point on the line segment

joining these points will also give the same

maximum (or minimum) two values

d. All of the above

17. The corner point method for bounded feasible

region comprises of the following steps

I. When the feasible region is bounded, M and

m are the maximum and minimum values of

Z.

II Find the feasible region of the linear

programming problem and determine its corner

points.

III.Evaluate the objective function Z = ax + by at

each corner point. Let Mand m respectively

be the largest and smallest values of these

points.

The correct order of these above steps is

a. III.I, II b. II, III, I

c. II, I, III d. I, III, II

Terminologies Related to the Solution of

LPP and Solution of LPP by Graphical

Method

18. Maximum value of 12x +3y subject to constraints

x > 0, y > 0, x + y < 5 and 3x + y < 9 is

a. 15 b. 36

c. 60 d. 40

19. The coordinate of the point at which minimum

value of z = 7x – 8y, subject to the conditions

x + y – 20 < 0, y > 5, x > 0, y < 9 is attained, is

a. (20, 0) b. (15, 5)

c. (0, 5) d. (0, 20)

20. The minimum value of the objective function

Z = 2x + 10y for linear constraints x > 0,

y > 0, x – y > 0, x – 5y < – 5 is

a. 10 b. 15

c. 12 d. 8

21. The minimum value of Z = 5x – 4y subject to

constraints x + y < 10, y < 4 ; x, y > 0 will be at

the point

a. (10,4) b. (–10,4)

c. (6,4) d. (0,4)

22. The optimal value of the objective function is

attained at the point is

a. given by intersection of inequations with axes

only

b. given by intersection of inequations with X-

axis only

c. given by corner points of the feasible region

d. None of the above

23. Variables of the objective function of the linear

programming problem are

a. zero b. zero or positive

c. negative d. zero or negative

24. A furniture dealer deals in only two items namely

tables and chairs he has Rs. 5000 to invest and

space to store atmost 60 pieces. A table cost is

Rs. 250 and a chair Rs. 60. He can sell a table at

a profit of Rs. 15. Assume that, he can sell all the

items that he produced. The number of constraints

in the problem are

a. 2 b. 3

c. 4 d. 1





25. The maximum value of Z = 10x + 6y subject to

constraints x > 0, y > 0, x + y < 12, 2x + y < 20

is

a. 72 b. 80

c. 104 d. 110

Direction (Q. Nos. 26-31) Afurniture dealer deals

in only two items- tables and chairs. He has Rs.

50000 to invest and has storage space of atmost

60 pieces. A table costs Rs. 2500 and a chair Rs.

500. He estimates that from the sale of one table,

he can make a profit of Rs. 250 and that from the

sale of one chair a profit of Rs. 75. He wants to

know how many tables and chairs he should buy

from the available money so as to maximise his

total profit, assuming that he can sell all the items

which he buys.

On the basis of the above information, answer

the following questions.

26. The above stated optimisation problem is an

example of

a. linear programming problem

b. functional programming problem

c. computer programming problem


27. Which of the following are constraints for the given

problem?

a. Maximum investment is ~50000 and storage

space is for maximum of 60 pieces

b. Minimum investment is ~50000 and storage

space is for minimum of 60 pieces

c. Maximum investment is f 50000 and storage

space is for minimum of 60 pieces


28. If he buys tables only, then his profit will be

a. Rs. 500 b. Rs. 5000

c. Rs. 7500 d. Rs. 2500

29. Suppose he chooses to buy chairs only. Then, his

maximum profit is

a. Rs. 4500 b. Rs. 7500

c. Rs. 4000 d. Rs. 5700

30. If he choose 10 tables and 50 chairs, then his

maximum profit is

a. Rs. 6200 b. Rs. 6500

c. Rs. 6250 d. Rs. 6520

31. If x is the number of tables and y be the number

of chairs, then the given problem, mathematically

reduces to

a. maximise Z = 250x + 75y

subject to the constraints

5x + y < 100

x + y < 60

x.y > 0

b. minimise Z = 250x + 75y


5x + y < 100

x + y < 60

x.y > 0

c. minimise Z = 250x + 75y


5x + y > 100

x + y > 60

x.y > 0

d. maximise Z = 250x + 75y


5x + y > 100

x + y > 60

x.y > 0

Direction (Q. Nos. 32-34) The feasible solution

for a LPP is shown as below:

Let Z = 3x – 4y be the objective function. Then,

32. Maximum of Z occurs at

a. (5,0) b. (6, 5)

c. (6,8) d. (4, 10)

33. Minimum of Z occurs at

a. (0,0) b. (0, 8)

c. (5,0) d. (4, 10)





34. Maximum value of Z + + Minimum value of Z)

is equal to

a. 13 b. 1

c. –13 d. –17

35. The maximum value of Z = 4x + 3y, if the feasible

region for an LPP is shown in following figure, is

a. 112 b. 100

c. 72 d. 110

36. The objective function of an LPP is

a. a constraint

b. a function to be optimised

c. a relation between the variables


37. The maximum value of P = x + 3y such that

2x + y < 20, x + 2y < 20, x > 0, y > 0 is

a. 10 b. 60

c. 30 d. None of these

Special Cases of LPP

38. The optimal value of the objective function is

attained "at the point

a. given by intersection of inequations with axes

only

b. given by intersection of inequations with X-

axis only

c. given by corner points of the feasible region


39. Maximum value of Z = 3x + 4y subject to

x - y > – t –x + y < 0; x, y > 0, is given by

a. 1 b. 4

c. 6 d. no feasible region

40. The area of the feasible region for the following

constraints 3y + x > 3,x M < 0, y > 0 will be

a. bounded b. unbounded

c. convex d. concave

41. The linear programming problem

Maximize Z = x1 + x

2

Subject to constraints

xl + 2x

2 < 2000

x1 + x

2 : < 1500

x2 < 600

x1 > 0 has

a. no feasible solution

b. unique optimal solution

c. a finite number of optimal solutions

d. infinite number of optimal solutions

42. Consider the linear programming problem

Maximise Z = 4x + y.

Subject to constraints x + y < 50, x + y > 100 and

x, y > 0. Then, maximum value of Z is

a. 0 b. 50

c. 100 d. Does not exist

43. Which of the following sets are not convex?

a. {(x,y) : 8x2 + 6y2 < 24}

b. {(x, y) : 6 < x2 + y2 < 36}

c. {(x,y) : y > 3,y > 30}

d. {(x,y) : x2 < y]

44. One of the important class of optimisation problem

is

a. functional programming problem

b. linear programming problem

c. numerical programming problem


45. The problems which seek to maximise (or

minimise) profit (or cost) from a general class of

problems called

a. optimisation problems

b. customisation problems

c. Both (a) and (b)

d. None of these

46. The wide applicability of linear programming

problem is in

a. industry b. commerce

c. management science d. All of these

47. An optimisation problem may involve finding

a. maximum profit

b. minimum cost

c. minimum use of resources

d. All of the above





48. If a young man rides his motorcycle at 25 km/h,

he has to spend Rs. 2 per km in petrol and if he

rides it at 40 km/h, the petrol cost rises to Rs. 5

per km. He has Rs. 100 to spend on petrol and

wishes to find the maximum distance, he can travel

within one hour. If x and y denote the distance

travelled by him (in km) at 25 km/h and 40 km/h,

respectively.

The inequations represent the data are

a. 2x + 5y < 100, x y

25 40 > 1, > 0, y > 0

b. 2x + 5y > 100, x y

25 40 > 1 x > 0, y > 0"

c. 2x + 5y < 100, x y

25 40 < 1,x > 0, y > 0

d. 2x + 5y < 100, 25x + 40y < 1 x > 0, y > 0

49. The maximum and minimum values of the

objective function Z = 3x–4y.

subject to the constraints,

x – 2y < 0

–3x + y < 4

x – y – 6

x.y > 0

are respectively

a. 12,10 b. 10,12

c. 12, does not exist d. 5,12

50. The maximum value of Z = 4x + 2y subject to

the constraints 2x + 3y < 18, x + y > 10

x, y > 0 is

a. 20

b. 36

c. 40

d. None of these

1. The point which provides the solution of the linear

programming problem, maximise Z = 45x + 55y.

Subject to constraints x, y > 0, 6x + 4y < 120 and

3x + 10y < 180, is

a. (15,10) b. (10,15)

c. (0, 18) d. (20, 0)

2. The maximum value of z = 10x + 6y, subject to

constraints x > 0, y > 0, x + y < 12, 2x + y < 20

is

a. 72 b. 80

c. 104 d. 110

3. If x + y < 2, x > 0, y > 0 is the point at which

maximum value of 3x + 2y attained will be

a. 72 b. (0, 0)

c. 104 d.1 1

,2 2

4. Maximum value of Z = 12x + 3y, subject to

constraints x > 0, y > 0, x + y < 5 and 3x + y < 9

is

a. 15 b. 36

c. 60 d. 40

5. Solve the linear programming problem.

Maximise Z = x + 2y

Subject to constraints x –y < 10, 2x+3y < 20 and

x > 0, y > 0

a. Z = 10 b. Z = 30

c. Z = 40 d. None of these

6. For an LPP, minimise Z = 2x + y subject to

constraints 5x + 10y < 50, x + y > 1, y < 4 and

x, y > 0, then Z is equal to

a. O b. 1

c. 2 d. 12

7. Consider the inequalities x1+x

2 < 3, 2x

1+5x

2> 10;

x1, x

2 2 > 0.Which of the point lies in the feasible

region?

a. (2,2) b. (1,2)

c. (2,1) d. (4,2)

8. Z = 4x + 2y, 4x + 2y 2: 46, x + 3y s 24 and x and

y are greater than or equal to zero, then the

maximum value of Z is

a. 46 b. 96

c. 52 d. None of these

9. The minimum value of z = 2x1 + 3x2 subjected

to the constraints

2x1+7x

2 22, x

1 + x

2 > 5x

1+x

2 > 10 and x

1,x

2 2y > 0

is

Exercise 2

(Miscellaneous Problems)





a. 14 b. 20

c. 10 d. 16

10. The maximum value of = 3x + 4y, subjected to

the conditions x + y < 40, x+2y < 60; x, y > 60, is

a. 130 b. 140

c. 40 d. 120

11. Consider the inequalities 5x1 + 4x

2 > 9, x

1+x

2 < 3,

x1 > 0, x

2 > 0. Which of the following point lies

inside the solution set?

a. (1, 3) b. (1, 2)

c. (1, 4) d. (2, 2)

12. The minimum and maximum values of Z for the

problem, minimise and maximise Z = 3x + 9 Y


x + 3y < 60

x + y > 10

x < y

x > 0, y > 0

are respectively

a. 60 and 180 b. 180 and 60

c. 50 and 190 d. 190 and 50

13. The linear programming problem minimise

Z = 3x+ 2y


x + y < 8

3x + 5y < 15

x > 0, y > 0 has

a. one solution

b. no feasible solution

c. two solutions

d. infinitely many solutions

14. The maximum and minimum values of the

objective function Z = x + 2y


x + 2y > 100

2x – y < 0

2x + y < 200

x,y 2 > 0

occurs respectively at

a. one point and three points

b. two points and one point

c. one point and infinitely points

d. one point and one point

15. The maximum value of the objective function

Z = 3x+4y


x + y < 4

x > 0, y > 0 is

a. 16 b. 18

c. 20 d. 25

16. Let x and yare the number of tables and chairs

respectively, on which a furniture dealer wants to

make profit for the constraints

Maximise Z = 250x + 75Y

5x + y < 100

x + y < 60

x > 0

y > 0

Consider the following graph

Then, the maximum profit to the dealer results

from buying

a. 10 tables and 50 chairs

b. 50 tables and 10 chairs

c. 0 table and 60 chairs

d. 20 tables and 40 chairs

17. The graphical solution of linear inequalities

x + y > 5 and x – y < 3, where L1 x + y = 5 and

L2 x– y = 3,is





18. By graphical method, the solutions of linear

programming problem maximise Z = 3xl + 5x

2

subject of contraints 3xl + 2x

2 < 18

x1 < 4, y

2 < 6 x

1 > 0, x

2 > 0 are

a. xl = 2, x

2 = 0, z = 6

b. x, = 2,x2 = 6, Z = 36

c. x1 = 4, x

2 = 3, z = 27

d. x1 = 4, x

2 = 6, z = 42

19. A toy manufacturer produces two types of dolls;

a basic version doll A and a deluxe version doll

B. Each doll of type B takes twice as long to

produce as one doll of type A.The company

have time to make a maximum of 2000 dolls of

type A per day, the supply of plastic is sufficient

to produce 1500 dolls per day and each type

requires equal amount of it. The deluxe version,

i.e. type B requires a fancy dress of which there

are only 600 per day available. If the company

makes a profit of Rs. 3 and Rs.5 per doll,

respectively, on doll A and B, then the number of

each should be produced per day in order to

maximise profit, is

a. 800, 500 b. 500, 600

c. 450,450 d. 1000,500

Direction (Q. Nos. 20-23) Consider the graph of

constraints stated as linear inequalities as below:

5x + y < 100 ... (i)

x+ y < 60 ... (ii)

x > 0 ... (iii)

y > 0 ... (iv)

where, x and yare number of tables and chairs on

which a furniture dealer wants to make his profit.

20. The shaded region in the graph is called

a. feasible region b. critical region

c. non-feasible region d. None of these

21. Every point in the feasible region is called

a. no solution b. unbounded solution

c. feasible solution d. non-feasible solution

22. There is/are ... D. .. point(s) in the feasible region

OABC that satisfy all the constraints given in

(i) to (ii). Here, D refers to

a. Only 1 b. Only 2

c. finitely many d. infinitely many

23. Any point in the feasible region that gives the

optimal value (maximum or minimum) of the

objective function is called alan

a. optimal solution b. no solution

c. finite solution d. None of these

24. Let R be the feasible region (convex polygon) for

a linear programming problem and Z = ax + by be

the objective function.

Then, which of the following statements is false?

a. When Z has an optimal value, where the

variables x and yare subject to constraints

described by linear inequalities, this optimal

value must occur at a corner point (vertex) of

the feasible region

b. If R is bounded, then the objective function

Z has both a maximum and a minimum value

on R and each of these occurs at a corner

point of R

c. If R is unbounded, then a maximum or a

minimum value of the objective function may

not exist

d. If R is unbounded and a maximum or a

minimum value of the objective function, then

it does not occur at a corner point

25. The minimum value of the objective function

Z = x + 2y

subject to the constraints,

2x + y > 3,

x + 2y > 6

x, y > 0 occurs

a. at every point on the line x + 2y = 6

b. at every point on the line 2x + y = 3

c. at every point on the line x + 2y = 3

d. at every point on the line 2x + y = 6

26. Let the feasible region of the linear programming

problem with the objective function Z = ax + by

is unbounded and let Mand m be the maximum

and minimum value of Z, respectively.

Now, consider the following statements





I. M is the maximum value of Z, if the open half

plane determined by ax + by > M has no point

in common with the feasible region. Otherwise,

Z has no maximum value.

II. m is the minimum value of Z, if the open half

plane determined by ax + by < m has no point

in common with the feasible region. Otherwise,

Z has no minimum value.

Choose the correct option.

a. Only I is true

b. Only II is true

c. Both I and II are true

d. Neither I nor II is true

27. The minimum and maximum values of the

objective function, Z = 5x + 10y


x + 2 < s 120, x + y > 60

x – 2y > 0, x, y > 0

are respectively

a. 300 and 500 b. 300 and 600

c. 600 and 700 d. 300 and 400

28. Consider the following statements

I. If the feasible region of an LPP is unbounded,

then maximum or minimum value of the

objective function Z = ax + by mayor may not

exist.

II. Maximum value of the objective function Z =

ax + by in an LPP always occurs at only one

corner point of the feasible region.

III.In an LPP, the minimum value of the objective

function Z = ax + by is always 0, if origin is

one of the corner point of the feasible region.

IV. In an LPP, the maximum value of the objective

function Z = ax + by is always finite.

Which of the following statements are true?

a. I and IV b. II and III

c. I and III d. II and IV

29. The corner points of the feasible region determined

by the system of linear constraints are (0, 10),

(5, 5) (15, 15), (0, 20). Let Z = px + qy , where

p, q > 0.

Then, the condition on p and q so that the

maximum of Z occurs at both the points (15, 15)

and (0, 20), is

a. p = q

b. p = 2q

c. q = 2p

d. q = 3 P

30. A cooperative society of farmers has 50 hectare

of land to grow two crops X and Y. The profit

from crops X and Y per hectare are estimated as

Rs. 10500 and Rs. 9000, respectively. To control

weeds, a liquid herbicide has to be used for crops

X and Yat rates of 20 Land 10 L per hectare.

Further no more than Rs.800 L of herbicide should

be used in order to protect fish and wildlife using

a pond which collects drainage from his lan

To maximise the total profit of the society, the

land should be allocated to each crop is

a. 20 hec for crop X and 30 hec for crop Y

b. 20 hec each for both crop X and crop Y

c. 30 hec for crop X and 20 hec for crop Y

d. 30 hec each for both crop X and crop Y

31. Anil wants to invest atmost Rs. 12000 in bonds

A and B. According to the rules, he has to invest

atieast Rs. 2000 in bond A and atleast Rs. 4000

in bond B. If the rate of interest in bond A is 8%

per annum and on bond B is 10% per annum,

then to maximise the interest, the investment in

bond A and B are respectively

a. Rs. 10000 and Rs. 2000

b. Rs.2000 and Rs.10000

c. Rs.6000 and Rs. 6000

d. None of these





MHT-CET Corner

1. The constraints – x1 + x

2 < 1,–x

1 + 3x

2 < 9,

x1,x

2 > 0 defines on

a. bounded feasible space

b. unbounded feasible space

c. both bounded and unbounded feasible space


2. The maximum value of the objective function

Z = 3x + 2y for linear constraints x + y < 7,

2x + 3y < 16, x > 0, y > 0 is

a. 16 b 21

c. 25 d. 28

3. The maximum value of z = 9 x + 13 y subject to

constraints 2x + 3 y – 18, 2x + y < 10, x > 0,

y > 0 is

a. 130 b. 81

c. 79 d. 99

4. For the LPP Min Z = x1 + x

2 such that inequalities

5x1 + 10x

2 > 0, x

1+ x

2 < 1, x

2 < 4 and x

1 x

2 > 0

a. There is a bounded solution 2008

b. There is no solution

c. There are infinite solutions


5. The region represented by the inequation system

x, y > 0, y < 6, x + y < 3, is

a. unbounded in first quadrant

b. unbounded in first and second quadrants

c. bounded in first quadrant


6. A wholesale merchant wants to start the business

of cereal with Rs. 24000. Wheat is Rs.400 per

quintal and rice is Rs. 600 per quintal. He has

capacity to store 200 quintal cereal. He earns the

profit Rs. 25 per quintal on wheat and Rs. 40 per

quintal on rice. If he stores x quintal rice and y

quintal wheat, then for maximum profit the

objective function is

a. 25x + 40y b. 40x + 25y

c. 400x + 600y d.400 600

x40 25

y

7. Which of the term is not used in a linear

programming problem?

a. Optimal solution b. Feasible solution

c. Concave region d. Objective function

8. If given constraints are 5x + 4y > 2, x < 6, y < 7,

then the maximum value of the function

Z = x + 2y is

a. 13 b. 14

c. 15 d. 20

9. Z = 30x + 20y, x + y < 8,x + 2y > 4, 6x + 4y > 12,

x > 0, y > 0 has

a. unique solution b. infinitely many solution

c. minimumat (4, 0) d. minimum60 at point (0, 3)

10. Minimize Z = 3x + y, subject to constraints

2x + 3y < 6, x + y > 1, x > 0, y > 0. Then

a. x = 1,y = 1 b. x = 0, y = 1

c. x = 1,y = 0 d. x = –1, y = –1

11. The shaded region for the inequality x + 5y < 6 is

a. to the non-origin side of x + 5y = 6

b. to the either side of x + 5y = 6

c. to the origin side of x + 5y = 6

d. to the neither side of x + 5y = 6





Answers

Exercise 1

1. (b) 2. (a) 3. (c) 4. (d) 5. (c) 6. (a 7. (b) 8. (c) 9. (d 10. (c)

11. (c) 12. (c) 13. (a) 14. (a) 15. (c) 16. (a 17. (b) 18. (b) 19. (d 20. (b)

21. (d) 22. (c) 23. (b) 24. (c) 25. (c) 26. (a 27. (a 28. (b) 29. (a 30. (c)

31. (a) 32. (a) 33. (b) 34. (d) 35. (a) 36. (b) 37. (c) 38. (c)) 39. (d 40. (b)

41. (d) 42. (d) 43. (b) 44. (b) 45. (a) 46. (a 47. (d 48. (c)) 49. (c)) 50. (d)

Exercise 2

1. (b) 2. (c) 3. (c)) 4. (b) 5. (d) 6. (b) 7. (b) 8. (b) 9. (a 10. (b)

11. (b) 12. (a) 13. (b) 14. (c) 15. (a) 16. (a 17. (c)) 18. (b) 19. (d 20. (a)

21. (c) 22. (d) 23. (a 24. (d 25. (a) 26. (c)) 27. (b) 28. (a 29. (d 30. (c)

31. (c)

MHT- CET Corner

1. (b) 2. (b) 3. (c) 4. (a) 5. (c) 6. (b) 7. (c)) 8. (d 9. (d 10. (b)

11. (c)




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